On the power of algebraic branching programs of width two

Allender, Eric; Wang, Fengming

doi:10.1007/s00037-015-0114-7

Cited by 14 publications

(32 citation statements)

References 9 publications

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“…Therefore, it is interesting to examine the power of 2-register noncommutative arithmetic computations. Width-2 noncommutative ABPs are also not universal [AW11]. However, the noncommutative version of Model 1 is universal.…”

Section: Noncommutative 2-register Arithmetic Computationsmentioning

confidence: 96%

“…Model 1 is universal by Lemma 1. In the absence of instruction (3) (which allows resets R i := c for any c ∈ F), the model can be simulated by width-2 arithmetic branching programs (ABP), which are not universal, as shown by Allender and Wang [AW11]. As mentioned earlier, they give a sparse polynomial that is not width-2 ABP computable.…”

Section: Epiloguementioning

confidence: 99%

“…Allender and Wang [AW11] have shown that 2-register ABPs (defined similarly via the iterated product of 2 × 2 matrices with affine linear form entries) are not even universal. Indeed, they show that the quadratic polynomial x 1 x 2 +x 2 x 3 +...+x n−1 x n cannot be computed by width-2 ABPs.…”

Section: Theoremmentioning

confidence: 99%

See 2 more Smart Citations

Computing and Combinatorics

2014

Lecture Notes in Computer Science

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Section: Noncommutative 2-register Arithmetic Computationsmentioning

confidence: 96%

Section: Epiloguementioning

confidence: 99%

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Computing and Combinatorics

2014

Lecture Notes in Computer Science

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“…As a matter of curiosity, one may want to know: is the width-3 upper bound tight? Allender and Wang [AW11] recently settled this question affirmatively: they show that a very simple polynomial cannot be computed by any width-2 ABP, no matter what the length. On the other hand, width-3 ABPs are universal, since every polynomial family has some formula family computing it.…”

Section: The Current Statusmentioning

confidence: 98%

Algebraic Complexity Classes

Mahajan

2014

Perspectives in Computational Complexity

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“…Our results show that border continuant complexity is polynomially equivalent to border formula size. This insight is quite striking because a result of Allender and Wang [1] implies that the continuant complexity without allowing approximations can be infinite! Continuous Lower Bounds.…”

Section: Introductionmentioning

confidence: 99%

On Algebraic Branching Programs of Small Width

Bringmann

Ikenmeyer

Zuiddam

2018

J. ACM

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In 1979, Valiant showed that the complexity class VP e of families with polynomially bounded formula size is contained in the class VP s of families that have algebraic branching programs (ABPs) of polynomially bounded size. Motivated by the problem of separating these classes, we study the topological closure VP e , i.e., the class of polynomials that can be approximated arbitrarily closely by polynomials in VP e. We describe VP e using the well-known continuant polynomial (in characteristic different from 2). Further understanding this polynomial seems to be a promising route to new formula size lower bounds. Our methods are rooted in the study of ABPs of small constant width. In 1992, Ben-Or and Cleve showed that formula size is polynomially equivalent to width-3 ABP size. We extend their result (in characteristic different from 2) by showing that approximate formula size is polynomially equivalent to approximate width-2 ABP size. This is surprising because in 2011 Allender and Wang gave explicit polynomials that cannot be computed by width-2 ABPs at all! The details of our construction lead to the aforementioned characterization of VP e. As a natural continuation of this work, we prove that the class VNP can be described as the class of families that admit a hypercube summation of polynomially bounded dimension over a product of polynomially many affine linear forms. This gives the first separations of algebraic complexity classes from their nondeterministic analogs. CCS Concepts: • Theory of computation → Algebraic complexity theory;

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On the power of algebraic branching programs of width two

Cited by 14 publications

References 9 publications

Computing and Combinatorics

Computing and Combinatorics

Algebraic Complexity Classes

On Algebraic Branching Programs of Small Width

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